The P-positions of the 2-pile take-away game of Wythoff Nim lie on two beams of slope √5+1 2 and √5?1 2 respectively. We study extensions to this game where a player may also remove simultaneously pt tokens from either of the piles and qt from the other, where p < q are given positive integers and where t ranges over the positive integers. We prove that for certain pairs (p, q) the P-positions are identical to those of Wythoff Nim, but for (p, q) = (1, 2) they do not even lie on two beams. By several experimental results, we conjecture a classification of all pairs (p, q) for which Wythoff Nim’s beams of P-positions transform via a certain splitting behavior, similar to that of going from 2-pile Nim to Wythoff Nim.
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